What's an infinite dimensional manifold and how can it be useful in hospitals

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What's an infinite dimensional manifold and how
can it be useful in hospitals?

• David Mumford, Brown University
• October 23, 2007
• Swarthmore College

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• There has been a huge explosion in medical
  imaging, in spatial and temporal resolution, in
  imaging many processes as well as organs
• New mathematical tools have entered the game,
  which are not so elementary.
• In particular, infinite-dimensional manifolds of
  shapes and infinite-dimensional groups of
  diffeomorphisms with associated Riemannian
  metrics have been used. This talk is an
  elementary step-by-step intro to what this is all
  about.

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Outline

• What are manifolds, charts and atlases?
• The set of closed plane curves S as an infinite
  dimensional manifold
• The 3D version and its application to anatomy
• 4. Exploratory data analysis and geodesics
• 5. BUT geodesics are not always act like
  straight lines curvature
• The outlook

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I. What is a manifold?
A manifold is a geometric thing which has local
charts, subsets where a flat set of coordinates
is given. But there are no global charts because
the whole manifold may be curved and may bend
back on itself e.g. take the surface of the
semi-circular canals in the ear. The set of
charts forms an atlas, whose charts cover the
manifold.
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The idea of an atlas some stuff off the web!
The abstract idea many pieces, on each have
coordinates x1,,xn
In dimension two, there are tori, pretzels,
surfaces with handles. Can (with some pain) make
an atlas for each.
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The 4 ingredients of differential geometry
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Some history

• All this (and more to come) is due to Gauss in
  the 2-D case.
• Riemann put together the n-D case in his
  Habilitation Lecture in 1854. This is not so well
  known, but here he also imagined the infinite
  dimensional version
• There are however manifolds in which the fixing
  of position requires not a finite number but
  either an infinite series or a continuous
  manifold of determinations of quantity. Such
  manifolds are constituted for example by the
  possible shapes of a figure in space, etc.

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II. The set S of all smooth plane curves forms a
manifold!
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An abstract view of what we are doing

• This whole blob represents the space of all plane
  curves
• Each curve represents a single point in the space
• The dotted lines represents parts which can be
  represented as deformations of the central shape
  forming a coordinate chart
• The sequence of shapes A,B,C,D,E are points along
  a curve in the space of shapes connecting a
  circle to a banana to a new moon.

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Think of S geometrically
or something with derivatives of a(s)
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The geometric heat equation defines a flow on
the manifold S

• In traditional terms, let Ct be yf(x,t). Then
  (after some work) it becomes
• which is the usual heat equation if the
  x-deriv of f is small.

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The geometric heat eqn is the gradient of curve
length!

• is a
  function on S
• To form a gradient, we need an inner prod
• The simplest inner product of 2 vectors
• What makes it work
• Move a curve normally proportional to its
  curvature this is the geometric heat eqn.
  
  

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III. The 3-D version and applications to anatomy

• All vertebrates are diffeomorphic (more or
  less) all healthy male (e.g. without tumors) and
  all healthy female humans are really clearly
  diffeomorphic with only moderate distortion.
• Can you, then, form an ideal 3D computer model
  of a male human and female human?
• THEN for each MRI or other scan of each
  patient, find an optimal diffeomorphism of the
  scanned region with the ideal model, revealing
  individual differences.

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A hippopotamus and a giraffe are indeed
diffeomorphic!
Matching by landmark points J. Glaunes
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What do we want to match up?

• The body has organs, bones, vessels with
  boundaries, well defined points (e.g. traditional
  points on the skull, like the nasion, menton or
  gonion), muscle fibre and nerve fibre tracts
  (giving line fields).
• The diffeomorphism should be constrained to
  respect these boundaries, points, orientations,
  maybe even some densities.

Kahler, Haber and Seidel, SIGGRAPH 2003
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• What is optimal? I
• The group G of diffeomorphisms is also a manifold
  and we can put a metric on it. Actually, consider
  the diffeomorphisms from the domain W of the scan
  to the domain W of the ideal human.
• Its tangent space is the linear space of vector
  fields on W and paths
  are described by
• We can consider all such diffeomorphisms which
  respect all these structures call this S, and
  consider the set of all such diffeomorphisms R
  which are rigid maps.

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• What is optimal? II
• Consider all paths in G from R to S. Take the inf
  of their lengths this measures the distortion.
  The endpoints of the path of least length is the
  optimal diffeomorphism.
• But what is length?
• Many choices for norm
• How many derivatives
• at least one is needed or
• metric collapses,
• Lp for some p. These
• choises have a huge effect
• on what optimal means.

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Some examples from the lab of M. Miller (JHU).
Left warpings of hearts, monkey cortex and
hippocampus. Below a non-diffeo with a tumor.
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Geodesics between faces Shortest paths in the
space of diffeos carrying one face to the other
(Vaillant, Trouve, Younes)
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Heart Mapping via Diffusion Tensor Magnetic
Resonance Imaging. The diffeo must respect the
fibre structure too.
Yan Cao, Winslow,Younes
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IV. Exploratory data analysis and geodesics
These are the standard work horses for data in
linear spaces. But what can we do for data on a
manifold? First, we need geodesics.
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What is a geodesic?
Navigating the earth, a shortest path is seldom a
straight line you must weave to avoid hills and
valleys. More generally, whenever a manifold has
an inner product defined on its tangent spaces,
one can seek the shortest path from A to B and
this is called a geodesic.
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Geodesics come from differential equations
Start with the variational principle
On a manifold with coordinates x1,,xn, get
Analogs on the infinite diml space of
diffeo-morphisms
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Data analysis via geodesics

• Given a dataset Pi on a manifold M, its
  Karcher mean is a point Q minimizing
• Once you have the mean, take the shortest
  geodesics from each Pi to Q and let
  be the tangent vector to this geodesic at Q.
• Then take the principal components via the linear
  theory on ti .
• k-means can also be done via Karcher means.
• This approach has been applied, e.g. to the shape
  of the hippocampus and the diagnosis of
  schizophrenia and Alzeimers to the shape of the
  heart in various conditions to the shape of the
  prostate etc.

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V. BUT geodesics are not always act like straight
lines curvature
The idea of curvature Euclid parallel lines
stay the same distance apart ZERO
CURVATURE Non-Euclidean geometry (Bolyai, Gauss)
geodesics diverge exponentially, e.g. at mountain
passes NEGATIVE CURVATURE Spherical geometry
great circles come together at antipodes similar
thing at mountain peaks or valleys. POSITIVE
CURVATURE
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Gravitational lensing positive curvature in
space-time. What you see is not what is out there!
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Curvature in infinite diml spaces
Geodesic triangles in the space of plane curves
(Michor-M metric) There is more than one way to
rotate an ellipse! For small shapes, curvature is
negative and the path nearly goes back to the
circle ( the origin). Angle sum 102
degrees. For large shapes, curvature is positive,
2 protrusions grow while 2 shrink. Angle sum
207 degrees.
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Geometry behind curvature

• Get curvature at each point in each 2-plane
  Riemanns sectional curvature and curvature
  tensor Rijkl, Ricci and scalar curvatures.
• When positive, beyond cut locus, geodesics are
  not unique. Datasets may not have means.
• When negative, easy to get lost, space is big
  but datasets do have means, geodesics are unique.

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VI. The outlook

• Curvature is a big obstacle to doing analysis and
  statistics on non-linear spaces. It affects
  discrimination and clustering, e.g. by fitting
  Gaussian models to data. But it reflects the
  non-linear nature of these big spaces shapes and
  diffeos dont live in vector spaces but have
  their own geometry.
• These infinite dimensional spaces have their own
  geometry which is only partially understood.
  Calculations can be made (Mario Miceli)
• A huge challenge is to define probability
  measures on S and G this is essential to e.g.
  maximum likelihood tests (is this shape a cat or
  a dog?).